Is it correct to use two present participles sequentially?

"We are currently in the process of finishing planning for the outage."

There can be grammatical constraints on some types of double "-ing" phrases.

I'll mention some of them here, and let you decide how applicable they are to your example.

Double -ing constraint:

CGEL page 1243-4:

The double-ing constraint

Some verbs that license gerund-participial complements cannot themselves occur in the gerund-participle form when they have such a complement. Compare:

[37]

• i a. They started quarreling. - - - b. (*) They are starting quarreling.

• ii a. The lawn needs mowing. - - - b. (*) The lawn is always needing mowing.

• iii a. We considered buying one. - - - b. We are considering buying one.

The succession of gerund-participles in [i.b/ii.b] is excluded by what is known as the 'double-ing constraint'. As evident from [iii.b], it applies to only a subset of catenative verbs -- a small subset, in fact. The clearest cases are aspectual verbs such as begin, cease, continue, start, stop, and verbs taking concealed passives, like need i [ii]. We should probably also include others, such as intend, but there is a good deal of variation between speakers as to their acceptance of the [b] construction. We noted in &1.1. that gerund-participials cannot have the progressive auxiliary be as head ( (*)They accused him of being running away when the alarm sounded), and this can be seen as a special case of the constraint.

Note that "finish" is an aspectual verb. CGEL page 1228:

Most aspectual verbs have raised subjects, . . . there are, however, a few that have ordinary subjects, normally with an agentive interpretation: discontinue, finish, quit, resume . . .

CGEL page 1174 fn1:

Examples with progressive be are occasionally encountered in casual speech: I've missed endless busses through [not being standing at the bus stop when they arrived]. This cannot, however, be regarded as an established construction in Present-day English.

It seems that you have got yourself a good ear. :)

Note that CGEL is the 2002 reference grammar by Huddleston and Pullum et al., The Cambridge Grammar of the English Language (CGEL).